12.009/18.352 Problem Set 3
Due Thursday, 6 March 2013
Problem 1: 40 pts — (a,b,c,d,e)=(5,10,10,5,10)
Problem 2: 30 pts — (a,b,c,d)=(10,10,5,5)
Problem 3: 30 pts — (a,b,c)=(15,10,5)
1. Equilibrium of CO2 with the Oceans & Timescale of CO2 Uptake: In class we discussed
how carbon dioxide (CO2 ) leaves a volcano and ends up both in the atmosphere and
in the ocean. If CO2 was not reactive in water then we could predict the concentration
of CO2 in the water based solely on its concentration in the air through our knowledge
of Henry’s Law (cough, 5.111), assuming that we waited long enough for the system
to reach equilibrium. Henry’s law normally takes the following form:
[CO2 ] = kH pCO2 .
(1)
Here, [CO2 ] is the concentration of carbon dioxide in the solution (units: M), kH is
the Henry’s law coefficient (a function of the chemical, solvent and temperature, units:
M atm−1 ) and pCO2 is the partial pressure of carbon dioxide in the gas phase (units:
atm; 1 atm=101325 Pa). This works quite well for finding the amount of dissolved
gas in a solution. Unfortunately for the purposes of this pset (though fortunately for
the habitability of Earth), carbon dioxide is not inert in water and can take on mul­
tiple forms. Two reactions in the ‘carbonate system’ and their respective equilibrium
constants are:
CO2 + H2 O
HCO−
3
+
+H ,
+
[HCO−
3 ][H ]
K1 =
[CO2 ][H2 O]
and
HCO−
3
+
CO2−
3 +H ,
K2 =
+
[CO2−
3 ][H ]
.
[HCO−
3]
(2)
(3)
In the following, we will assume the oceans have a known pH, where you should recall
that:
pH = −log10 [H+ ]
(4)
Because carbon dioxide takes part in these reversible reactions the concentration of
carbon dioxide stored in the ocean is not [CO2 ], but rather:
2−
[DIC] = [CO2 ] + [HCO−
3 ] + [CO3 ],
(5)
where [DIC] signifies “dissolved inorganic carbon,” (units: M).
(a) Show that the expression for [DIC] as a function of the carbon dioxide concentra­
tion, the various equilibrium constants and the hydrogen ion concentration [H+ ]
1
(i.e. pH), is given by:
∗
[DIC] = kH
pCO2
K1
K1 K 2
∗
kH ≡ kH 1 + + + + 2
[H ]
[H ]
(6)
(b) Now we will use equation (6) to estimate the partitioning of carbon between the
atmosphere and ocean. Assume that the atmosphere has mass MA (units: kg),
mean molecular weight m̄ (units: kg/mol), a surface pressure of 1 atm, and that
CO2 is a trace species (has minimal impact on overall atmospheric mass, pressure,
etc.). Also assume that the ocean has volume VO (units: L). Now, derive an
expression for the mass fractions of carbon in the atmosphere and ocean – i.e.
how much carbon of a total amount CT is found in the ocean as [DIC], and how
much is found in the atmosphere as gaseous CO2 ? Explain qualitatively how the
ocean pH affects these mass fractions.
(c) At 275 K (i.e. close to the temperature of most of the world’s ocean water) kH =
6.7×10−2 M atm−1 , K1 = 3.3×10−7 M, and K2 = 2.9×10−11 M (kH , K1 , and K2
are all fairly temperature-dependent). Look up the mass of Earth’s atmosphere,
the volume of the ocean, and the average ocean pH, and estimate the partitioning
of carbon between the atmosphere and ocean. For a surface partial pressure
pCO2 =280 µ-atm, what is the mass of carbon in the atmosphere and ocean?
(d) The molecular diffusivity Dm of all of the components of [DIC] in water is on the
order of 10−9 m2 s−1 . Using this, and a characteristic ocean depth, derive a time
scale for molecular diffusion of carbon from the atmosphere to the bottom of the
ocean.
(e) You should have found a timescale estimate that is quite a bit too large. This
is for several reasons, most notably the fact that the ocean is a highly turbulent
fluid, allowing for water masses that were originally far from one another to be put
in close proximity, and greatly enhancing the surface area over which molecular
diffusivity can act. Typical estimates of the ocean’s vertical “eddy diffusivity,”
(De,v ) or effective diffusivity for a highly turbulent fluid, are De,v ∼ 5 × 10−5
m2 s−1 . Thus, diffusion is accelerated by many orders of magnitude due to the
interaction of turbulent mixing and molecular diffusion. Consider the surface
ocean layer of depth H which contains as much carbon as the entire atmosphere.
What is the timescale for eddy diffusion to this depth and how does it relate to
what we have discussed in class? What is the timescale for equilibration of the
ocean [DIC] following the addition of a pulse of CO2 to the atmosphere? Roughly
how much of the CO2 in a pulse would stay in the atmosphere for on very long
timescales?
2. Isotopic Keeling Curve: In class we discussed the famous Keeling curve which shows
how carbon dioxide levels in the atmosphere are not only increasing but oscillat­
ing at the same time. We’ve placed carbon dioxide data from three stations, Point
Barrow, Alaska; Mauna Loa, Hawaii; and South Pole Antarctica on the class web­
2
site. This data is included in a .mat file and contains three structures (PointBarrow,
MaunaLoa, and SouthPole). To give you a sense of how the data is actually structured
SouthPole.CO2.t is the year that the concentration data was taken and SouthPole.CO2.v
is the data in ppm. The carbon-13 data is in the same form with SouthPole.d13.t
being the time and SouthPole.d13.v the δ 13 C of atmospheric CO2 .
(a) Provide two different plots. On the first plot show how the carbon concentration
varies in time for all three stations. On the second plot show how the stable
isotope values change in time for all three stations.
(b) What causes the carbon dioxide concentration to oscillate? In what way does the
oscillation of the carbon isotope value support or refute your rationale?
(c) In what ways do the amplitude and phase (i.e. relative timing) of the oscillations
in both plots change between sampling cites? Provide a rational argument for
why these changes might occur. If you get stuck, look at where these sites are on
the globe for inspiration.
(d) Despite changes in the oscillations between each site, the general trend seems to
be fairly constant. What is the general trend for concentration and isotopic value
and why do you think it remains consistent between sites?
Š
3. Photons in the Sun: MATLAB Problem
This problem was inspired by an Encyclopedia Britannica article. We seek to answer
the question: How long does it take for a photon generated by the fusion of hydrogen
in the middle of the sun take to escape into the solar system? On the surface this
seems like a straightforward question. All we need to do is divide the radius of the sun
(7×108 meters) by the speed of light (3×108 meters/second) to get ∼2.6 seconds. This
would be correct except for the fact that a photon travels approximately 0.1 cm before
it is scattered in a random direction by an electron. We will approach this problem by
conducting a numerical experiment.
(a) Write a short MATLAB script to calculate the number of 0.1 cm steps a photon
will take before it travels a given distance from the center of the Sun. Please
include a paragraph explaining the logical steps your code takes to simulate this
process. (Note: There are many ways of coding this. I am not asking for a
masterpiece, only something that works. It is important to realize that because
this is an experiment you will need multiple photons at each distance to gain an
understanding of how this works)
(b) Make a log-log plot of the average number of steps required for a photon to reach
the following distances (1, 2, 3, 4, 5, 10, and 20 cm). Start with ∼10 photons
for each distance. Do you observe a functional relationship between the number
of steps and the distance traveled? Use MATLAB (e.g., polyfit) to fit the
parameters of the function. What is it? Can you explain where this relationship
comes from?
3
(c) Using the functional relationship you found with the aid of the numerical experi­
ment, estimate the average time it will take a photon generated in the center of
the sun to reach the Sun’s surface.
4
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12.009J / 18.352J Theoretical Environmental Analysis
Spring 2015
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